This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson–Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay–Rudin–Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.

中文翻译：

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非周期性量子系统的光谱连续性：民俗定理的应用

这项工作为符号动力学系统接受Hausdorff拓扑中的一系列周期逼近提供了必要和充分的条件。在这里证明和应用的关键结果使用了称为De Bruijn图，Rauzy图或Anderson-Putnam复数的图，具体取决于社区。将其与先前的结果相结合，本工作严格证明了用于数值计算大型自伴算子的谱的算法方法的准确性和可靠性。所谓的哈密顿量描述了非周期性介质中量子粒子的有效动力学。除了一般规律性假设外，不对这些运算符的结构施加任何限制。特别地，不需要最近邻相关。斐波那契数列和Golay–Rudin–Shapiro序列的示例已明确提供，以说明此讨论。尽管物理学家和数学家都对第一个序列进行了彻底的研究，但是在光谱特性方面，仍然笼罩着神秘的阴影。有鉴于此，本文在此处给出了新的结果，可能有助于发现解决方案。